#201632

order	512
nilpotency class	3
isoclinism	isoclinic class of #201414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#35190	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1227 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,5,9,6,10,4,8)(3,11,14,15,12,16,13,7)(17,18)(19,21)(20,24,30,32)(22,25)(23,26)(27,31,28,29);(1,3,6,12)(2,7)(4,13,5,14)(8,16)(9,11)(10,15)(17,19,21,18,23,25,22,26)(20,28,29,32,30,27,31,24);(3,12)(7,15)(11,16)(13,14)(17,20)(18,24)(19,27)(21,29)(22,31)(23,30)(25,28)(26,32);(1,4,6,5)(2,8,10,9)(3,13,12,14)(7,16,15,11)(17,21,23,22)(18,25,26,19)(20,29,30,31)(24,28,32,27)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=h,e^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ch,f^b=fi,e^c=ei,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317747, #72318850, #72318856
maximal normal subgroup of	#72317747, #72318850, #72318856

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108828);